There's a difference between a big league contract and the league minimum salary.
A player may sign a big league contract but not be on the big league club. The big league contract means he must be on the 40-man roster and, therefore, be paid the 40-man roster minimum of $54,500. Typically, this is the type of deal that some of the premier amateur draft prospects will sign. It provides them with some leverage, in that their option clock starts ticking sooner and they should reach the majors quicker.
If he's promoted to the big leagues, then his salary increases to the big league minimum (currently $327,000).